Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> imaginatorium@xxxxxxxxxxxxx said:
> >
> > aeo6 Tony Orlow wrote:
> > > imaginatorium@xxxxxxxxxxxxx said:
> > > >
> > > > aeo6 Tony Orlow wrote:
> > > > > mueckenh@xxxxxxxxxxxxxxxxx said:
> > > > > >
> > > > > >
> > > > > > Remember: Every set of even numbers contains numbers which
are
> > > > larger
> > > > > > tan the cardinality of that set. This is valid for *every*
set
> > of
> > > > > > finite numbers. There is no reasonable arguing claiming
that
> > this
> > > > > > situation should change in case of an "infinite" set.
> > > >
> > > > Yes there is - it's the error contained in the "salient point"
> > below.
> > > >
> > > > Also some
> > > > > > elements of an infinite set of even numbers would
necessarily
> > > > surpass
> > > > > > the cardinality.
> > > > > Yes. The **salient point** is that the set size can never be
> > LARGER than
> > > > the
> > > > > largest element.
> > > >
> > > > Absolutely! The set size can never be LARGER than the largest
> > element.
> > > > Shout it from the rooftops! If you can find the largest
element,
> > the
> > > > set size cannot be larger. Great! Do we all agree now??
> > > >
> > > > Uh-huh. Problem. What happens when there is NO largest element?
> > Then
> > > > the set size cannot be larger than, er, um, well, come in
Houston,
> > we
> > > > don't seem to have a largest element here.
> > > >
> > > > Brian Chandler
> > > > http://imaginatorium.org
> > > >
> > > >
> > > Brian I already responded to that, you fool.
> >
> > Insults are very unproductive. You do not help by throwing words
like
> > "moron" at people like Randy Poe, who show unimaginable patience in
> > pointing out your mistakes over and over again. Anyway...
>
> Excuse me, but he did not even read what I wrote. I keep having to
repeat
> myself as the cantorians change the subject and refuse to follow the
logic.
> People want top harp on the fact that there is no largest finite?
Fine. I
> changed it to "some element must be at least as large as the set
size." When
> people ask the same dumb questions and don't pay attention to
answers, or say
> you said something you didn't, just to avoid the point, what should I
say?

You could start by addressing the obvious logical problem:
T: "The largest element must be xyz..."
E: "There isn't a largest element"
T: "Oh, right, well, _something_ must be xyz..."

This simply doesn't follow. All your arguments about what happens with
a largest element are true, and you stress how significant this is
("salient point", you said); when there isn't a largest element, you
will have to start again to build a new argument. It simply won't do to
just hope you can use the old one when its salient point has gone.


> > > You claim an infinite set
> > > containing only finite values. You claim the argument fails
because
> > there is no
> > > largest element, but whatever it is, it's finite. Well, if all
values
> > in the
> > > set are finite, then they are all less than infinite, **so** the
set
> > size cannot be
> > > infinite. QED
> >
> > Why "so" (my asterisked emphasis)? The set consists entirely of
finite
> > values. The fact that there is an unlimited number of them does not
*of
> > itself* imply that any of these values is itself finite - as you
> > yourself agree, in the case of (e.g.) the rationals in [0, 1]. So
why
> > do you think it does in this case? Well, look at _your own_
argument.
> > The "salient point" you say is something to do with the largest
> > element; but then when we point out that no largest element exists,
> > somehow this "salient point" doesn't matter any more?
>
> When you wrote this, I had already posted two proofs.

Sorry, there are huge numbers of posts above with your name on. Please
repost (or send me a google link). In general all such "proofs" can be
regressed (ha ha!) to a quantifier swap. Your "largest element" above
is (a particular case of) an upper bound for the values of the set
elements. More generally, to say that there is an upper bound U means:

(1) Exists U, s.t. forall p, p an element of the set, then U >= p.

But all you can say *within* the natural numbers (N), is:

(2) Forall n a natural number, exists Q s.t. Q >= n (stronger: Q > n
if you like)

The fundamental difference between (1) and (2) is the order of
universal and existential quantifiers. It really helps if you have that
distinction clear in your head - one of the problems is that in natural
language we tend to be very cavalier about the use of words like 'all'
and 'every', and it's easy to slip up interpreting unfamiliar
statements.

<snip>

> > One of the
> > most basic things set theory allows us to do is to consider the
element
> > of all elements (within some universe of discourse) having a
particular
> > property.
> Set of all sets, you mean? No wait, let me try that again, in typical
math-
> group-speak:
>
> The element of all elements??

Sorry about the horrendous typo. (I did post a correction) I mean "set
of all elements..."


** Statement 11 **
> > When we say "The set of all x having property P", we mean
> > (obviously) that every single element x of this set does have the
> > property P.

> > ... Is this clear to you?

> Sure. It doesn't mean the statement is true.

So I can't be sure if I have led you astray with my confusion above.
But if you are saying that statement 11 above may not be true, you
really have misunderstood. Are you really saying that ** The set of all
x having property P ** may also include elements that do not have the
property P??

> > If so, when I say "The set of all integers x, each x being a finite
> > integer", why do you claim that by some mysterious process at least
one
> > other element y, not having the property of y being a finite
integer
> > will somehow creep in?
>
> All I have been saying, over and over and over again, is that if the
values are
> all finite, then the set size is necessarily finite, because with the
addition
> of an infinite number of values, each 1 greater than the last, the
values of
> the elements will achieve infinite values. Define your set as you
like, just
> keep in mind the implications of your definition.
>
> >
> > Several people have asked you - over and over again - well,
consider
> > the set with all the 'naughty' elements removed. In terms of your
> > "integers" (actually the 10-adics) including "numbers" with an
> > unlimited number of nonzero digits, do you think that if you
consider
> > any one of these it must either have or not have an unlimited
number of
> > nonzero digits? If so, how many of then do *not* have an unlimited
> > number of nonzero digits?
>
> Is that your definition of a finite number? Its incorrect.
100000..... has only
> one non-zero digit and is infinite.

"1000..." isn't even a 10-adic. 10-adics do have reasonable arithmetic:
you can add them, for example. Have you thought how you would add
100... and ...111 ?? Beware the slippery slope here: Defenders of the
Infinite Integers do tend to end up with ever increasingly ill-defined
patterns of digits and dots.

I had made a suggestion before that a zero
> in the leftmost[1] digit means its finite, but I see now that's not
right either.

[1] There isn't a leftmost digit

> A finite number is one where all nonzero digits are a finite number
of digits
> from the rightmost digit.

But OK, in the "Any-old-pattern-of-digits-and-dots" system, your
definition of finite will do.

> > > You certainly draw a lot of conclusions about a set which you
cannot
> > enumerate.
> >
> > Which set (precisely) is that?
>
> Gee, which set have we been discussing? I seem to remember one where
we can
> count forever and never stop, but the values are all finite, but
there are an
> infinite number of them, although every time you count and add the
next higher
> value to the set you also increase the set size by one, so even
though they
> grow at precisely the same rate, the set size achieves infinity,

You keep using this word "achieve". Unending processes do not
"achieve". Anyway, the problem is that you are discussing at least two
sets: the set of (redundantly: finite) natural numbers, and various
extensions thereof.

.

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